#7737511

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#45243	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4711 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

(9,10,13,12)(11,15,16,14)(17,18,23,27)(19,26,21,24)(20,25,31,22)(28,29,30,32);(1,2)(3,8)(4,5)(6,7)(9,11,13,16)(10,14,12,15)(17,19,23,21)(18,24,27,26)(20,28,31,30)(22,29,25,32);(2,7)(3,4)(17,20,23,31)(18,22,27,25)(19,28,21,30)(24,29,26,32);(1,3,6,4)(2,8,7,5)(9,10,13,12)(11,14,16,15)(17,20)(18,25)(19,28)(21,30)(22,27)(23,31)(24,32)(26,29);(1,4,6,3)(2,5,7,8)(9,10,13,12)(11,14,16,15)(17,21,23,19)(18,26,27,24)(20,30,31,28)(22,32,25,29);(1,5)(2,3)(4,7)(6,8)(9,12,13,10)(11,15,16,14)(17,22,23,25)(18,20,27,31)(19,29,21,32)(24,28,26,30)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7644707);